Gram-Schmidt Q: Orthogonalizing v1 & v2 -Help Appreciated!

[astropi]

Homework Statement

First off, this isn't for a class, I'm just going over some material, however this does come from a textbook, so I figure this is a reasonable place to ask the question! Here's the question:

Use the Gram-Schmidt procedure to orthogonalize the following vectors:

v1=[(1+i),1,i]
v2=[i,3,1]
v3=[0,28,0]

Homework Equations

Let's not even worry about v3 right now. Let's just orthogonalize v1 and v2.

The Attempt at a Solution

First off, we let v1=u1 = [(1+i),1,i]

Now, we can find u2 by: $$u2 = v2 - \frac{<u1,v2>}{||u1||^2}u1$$

The norm of u1 is 2, therefore squaring that we get 4.
When I took <u1,v2> I got 4. Therefore 4/4 = 1.
This leaves us with u2 = v2 - u1 = (-1,2,1-i)
HOWEVER, u2 dot u1 = 2
and of course if they were orthogonal they should equal 0.
Not sure where I made a mistake... so if anyone can help that would be appreciated!

cheers,

-astropi

 

[Dick]

(-1,2,1-i) dot ((1+i),1,i) is equal to zero. Not 2. I think you forgot a complex conjugate when you did the inner product.

 

[astropi]

(-1,2,1-i) dot ((1+i),1,i) is equal to zero. Not 2. I think you forgot a complex conjugate when you did the inner product.

Yes, indeed...
my mind has been meandering around Hilbert space too long ;)
Thanks!

 

[diazona]

Yes, indeed...
my mind has been meandering around Hilbert space too long ;)

Maybe you can be my tour guide :rofl: